1,2

According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe, Apr 08 2004

Note that n+1 must be prime for all n in this sequence. - T. D. Noe, Apr 28 2004

Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n<m+1: x=n, y=n(n-1), z=m+1-n. If n=m+1, n>3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0<k<m+1 : x=k, y=m+1-k, z=m(n^2-k^2)/(m+1)+k(k-1). - Herm Jan Brascamp (brashoek(AT)hi.nl), May 28 2007

Zhu and Shao computed the first 18 positive integers d for which there does not exist a positive definite indecomposable binary quadratic form over Z with discriminant d. Peters proved that the sequence of such integers is identical to this sequence. - Robin Visser, Oct 12 2025

Fu Zu Zhu and You Yu Shao, On the construction of indecomposable positive definite quadratic forms over Z, Chinese Ann. Math. Ser. B 9 (1988), no. 1, 79-94.

Jonathan Borwein and Kwok-Kwong Stephen Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.

Al-Zaid Hassan, B. Brindza, and Á. Pintér, On positive integer solutions of the equation xy+yz+xz=n, Canad. Math. Bull. 39 (1996), no. 2, 199-202.

Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.

Meinhard Peters, Indecomposable binary quadratic forms, Arch. Math. (Basel) 57 (1991), no. 5, 467-468.

Meinhard Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.

Fu-Zu Zhu, On the construction of indecomposable positive definite Hermitian forms over imaginary quadratic fields, J. Number Theory 62 (1997), no. 2, 353-367. See Table III on page 363.

n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]

(SageMath)

def is_A025052(k):

for (c, d) in [(x, y) for x in range(1, k//2+1) for y in ZZ(k+x^2).divisors()]:

if (d > c) and ((k+c^2)/d > c): return False

return True

print([k for k in range(1, 1000) if is_A025052(k)]) # Robin Visser, Oct 12 2025

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0<a<b<c) and of A006093.

Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.

Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0<a<b<c), A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

Sequence in context: A018164 A321403 A340311 * A142584 A393315 A098197

Adjacent sequences: A025049 A025050 A025051 * A025053 A025054 A025055

nonn,fini,nice

Corrected by R. H. Hardin

approved